Writing Dynamics in State Space Form Robert Platt Northeastern University
Motivation In order to reason about complex dynamical systems, we need to write system dynamics in a convenient form. How encode dynamics of an inverted pendulum? How plan walking trajectories? How plan flying trajectories?
A simple system k m b Force exerted by the spring: Force exerted by the damper: Force exerted by the inertia of the mass:
A simple system k m b Consider the motion of the mass • there are no other forces acting on the mass • therefore, the equation of motion is the sum of the forces: This is called a linear system. Why?
A simple system k Let's express this in ''state space form'': m b
A simple system k Let's express this in ''state space form'': m b
A simple system k Let's express this in ''state space form'': m b
A simple system k m Let's express this in ''state space form'': b
A simple system k m Let's express this in ''state space form'': b where
A simple system k f Your finger m b Suppose that you apply a force:
A simple system Suppose that you apply a force:
A simple system Suppose that you apply a force: Canonical form for a linear system
Continuous time vs discrete time Continuous time Discrete time
Continuous time vs discrete time Continuous time Discrete time What are A and B now?
Continuous time vs discrete time Continuous time Discrete time What are A and B now?
Simple system in discrete time We want something in this form:
Simple system in discrete time We want something in this form:
Simple system in discrete time We want something in this form:
Simple system in discrete time We want something in this form:
Continuous time vs discrete time CT DT CT DT
Continuous time vs discrete time CT DT CT DT In this class, we’re going to focus on discrete time representations...
Think-pair-share External force Viscous damping Express DT dynamics of this system in state space form
Think-pair-share Express DT dynamics of this system in state space form
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